On a class of arithmetical convolutions

An algebraic independence result for Euler products of finite degree

“…Alkan and the authors [1] generalized this construction and provided a family of extensions of A r which are discrete valuation rings. For other work on rings of arithmetical functions the reader is referred to [5], [6], [9], [12], [13], [10], [11], [2]. In [1], it was shown that for any completely additive arithmetical function ψ ∈ A r , the map D ψ : A r → A r defined by D ψ (f )(n 1 , .…”

On a class of $\psi$-convolutions characterized by the identical equation

“…In [7] the identical equation (1.1) has been generalized to a class of 0-convolutions (which are certain binary operations in the set of arithmetic functions introduced by D.H. Lehmer [2]): if (m, n) E T, where T is the domain of 1/1, then for any multiplicative function f, f -1 being the inverse of f with respect to 0 (Section 2 contains undefined notions used in this section); this class of 0-convolutions in which (1.4) has been established is contained in the class of 0-convolutions preserving multiplicativity and satisfying ~(x, y) > max{x, y} for x, y E T, as subclass: in addition to these, they satisfy certain other properties similar to those of regular A-convolutions [3]. It is interesting to note that these convolutions have been subsequently characterized (see [8], Corollary 4.1; also see §2 of the present paper) and have been named [8] as Lehmer-Narkiewicz convolutions.…”

“…The statements " (X 7 Y) E T, (~(~, y), z) E T" (2.3) and "(y, z) E T, (x, V) (y, z)) E T" are equivalent; if one of these I conditions holds, we have ~(~(x, y), z) _ ~(~, ~(y, z)). Let For each positive integer n, if A(n) is a non-empty subset of divisors of ~, Narkiewicz [3] It may be noted that the multiplicativity preserving property of '0 is not a necessary condition for the validity of (1.4). For example, if T = 1 (1, n), (n, 1) : n E and = n for all n E Z+, then (1.4) holds trivially for all arithmetic functions f with f (1) = 1.…”

An arithmetical equation with respect to regular convolutions

“…For each positive integer n let A(n) be a subset of the set of positive divisors of n. Then the A-convolution [11] of two arithmetical functions f and g is defined by…”

“…An A-convolution is said to be regular [11] if (i) the set of arithmetical functions forms a commutative ring with identity with respect to the usual addition and the A-convolution,…”